Alternatively, in cases where the Dirac pairing \(\langle -,-\rangle \)  is degenerate (as is the case for local CY threefolds) such that there exists a non-zero \(\delta \in \Gamma \)  with \(\langle \delta ,-\rangle =0\) , one may perturb the charges \(\gamma _i \rightarrow \gamma _i + \epsilon _i \delta \)  where
and \(\epsilon _i\)  are small parameters, such that the attractor flow tree is resolved into a union of binary trees, for which (REF ) or (REF ) can be applied. Such a perturbation is in fact a
necessary step when applying the flow tree formula [1]}, [2]} at each vertex, although it need not extend to a global perturbation of the full set of trees in
\(\mathcal {T}_z(\lbrace \gamma _i\rbrace )\) .
